# Excercise 5.2 Lines-and-Angles - NCERT Solutions Class 7

Exercise 5.2

## Question 1

State the property that is used in each of the following statements?

(i) If \(a || b,\) then \(\angle1 = \angle5.\)

(ii) If \(\angle4 = \angle6,\) then \(a || b.\)

(iii) If \(\angle4 + \angle5 = 180^\circ,\) then \(a || b.\)

### Solution

**Video Solution**

**Steps:**

(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are equal .

(ii) When a transversal intersects two parallel lines such that if pair of alternate interior angles are equal then the lines are parallel.

(iii) When a transversal intersects two parallel line such that pair of interior angles on the same side of transversal are supplementary, then the lines are parallel.

## Question 2

In the adjoining figure, identify:

(i) The pairs of corresponding angles.

(ii) The pairs of alternate interior angles.

(iii) Pairs of interior angles on the same side of the transversal.

(iv) The vertically opposite angles.

### Solution

**Video Solution**

**Steps:**

(i) \(\angle1\) and \(\angle5\); and \(\angle2\) and \(\angle6\);\(\angle4\) and \(\angle8\);\(\angle3\) and \(\angle7\);

(ii) \(\angle3\rm \,and\,\angle5 \,, \angle2 \,and\,\angle8\)

(iii) \(\angle3\,{\text {and} }\,\angle8,\angle2\,{\text {and} }\,\angle5\)

(iv) \(\angle1\) and \(\angle3\); \(\angle2\) and \(\angle4\); \(\angle6\) and \(\angle8\); \(\angle5\) and \(\angle7\)

## Question 3

In the adjoining figure, \(p || q\). Find the unknown angles

### Solution

**Video Solution**

**Reasoning:**

First, by using linear pair find the measure of \(\angle e\) and then find the corresponding and vertically opposite angle to \(\angle e\) and again by using linear pair find the value of \(\angle b\) and its vertically opposite angle.

**Steps:**

Given \(p || q\) and it is intersected by a transversal.

\(\angle = 125^\circ\\ \text{(Corresponding angle)}\)

Since,

\(125^\circ+ \angle e = 180^\circ \\ \text{(Linear pair)}\)

\(\angle e = 180^\circ - 125^\circ\)

\(\angle e = 55^\circ\)

\(\angle e = \angle f = 55^\circ \\ \text{(Vertically opposite angles)}\)

\(\angle e = \angle a = 55^\circ \\ \text{(Corresponding angles)}\)

\(\angle a + \angle b = 180^\circ \\ \text{ (Linear pair)}\)

\(55^\circ + \angle b = 180^\circ\)

\(\angle b = 180^\circ - 55^\circ\)

\(\angle b =125^\circ\)

Also,

\(\angle b = \angle d = 125^\circ \\ \text{(Vertically opposite angles)}\)

\(\angle a = \angle c = 55^\circ \\ \text{(Vertically opposite angles)}\)

Thus,

\(\angle a = 55^\circ; \angle b = 125^\circ; \angle c = 55^\circ;\)

\( \angle d = 125^\circ;\angle e = 55^\circ ; \angle f = 55^\circ\)

## Question 4

Find the value of \(x\) in each of the following figures if \(l || m.\)

### Solution

**Video Solution**

**(i) Reasoning:**

There are two operations done in sequence. First, find the corresponding angle to \(110^\circ\) i. e \(\angle y,\) then by using Linear pair find the value of \(\angle x.\)

According to this model, the result \(\angle x + \angle y = 180^\circ.\) Now, it’s a matter of finding value of \(\angle x.\)

**(i) Steps:**

Solve for \(x\)

Given \(l || m\) and \(t\) is transversal,

\[\begin{align}\angle y &= 110^\circ\text{(Corresponding angle)}\\ \angle x+\angle y &= 180^\circ\text{(Linear pair)}\\\angle x &= 180^\circ - 110^\circ\\\angle x &= 70^\circ\end{align}\]

**(ii) Reasoning:**

Let’s visually model this problem. There is one operation that can be done. Find the corresponding angle to \(x.\) According to this model, the resultant value of corresponding angle will be equal to \(x.\) Now, it’s a matter of finding measure of \(x.\)

**(ii) Steps:**

Solve for \(x\)

Given \(l || m\) and \(a || b,\)

\[\angle x = 100^\circ \text{(corresponding angle)}\]

## Question 5

In the given figure, the arms of two angles are parallel. If \(\angle ABC = 70^\circ,\) then find:

(i) \(\angle DGC\)

(ii) \(\angle DEF\)

### Solution

**Video Solution**

**Reasoning:**

Let’s visually model this problem. There is one operation that can be done. According to this model, the resultant value of corresponding angle will be equal to \(\angle DGC.\)

Now, it’s a matter of finding measure of \(\angle DGC.\)

**Steps:**

(i) Solve for \(\angle DGC\)

Given \(AB || DG\) and \(BC\) is transversal

Also,

\(\angle ABC = 70^\circ\text{ (Given)}\)

Since,

\(\angle ABC = \angle DGC\text{(Corresponding angles)}\)

Therefore\(, \angle DGC = 70^\circ→(1)\)

**Reasoning:**

Let’s visually model this problem . There is one operation that can be done.According to this model, the resultant value of corresponding angle will be equal to \(\angle DGC.\) Now, it’s a matter of finding measure of \(\angle DEF.\)

**Steps:**

(ii) Solve for \(\angle DEF\)

Given \(BC || EF\) and \(DE\) is transversal

Also, \(\angle DGC = 70^\circ \text{(from 1)}\)

Since,

\(\angle DGC = \angle DEF \text{(Corresponding angles)}\)

Therefore\(, \angle DEF = 70^\circ\)

## Question 6

In the given figures below, decide whether \(l\) is parallel to \(m.\)

### Solution

**Video Solution**

**(i)** **Reasoning:**

Let’s visually model this problem. There is one operation that can be done check whether interior angles are supplementary or not. According to this model, the result sum of \(126^\circ + 44^\circ\) is \(170^\circ.\) Now, it’s a matter of finding \(l\) is parallel to \(m\) or not

**Steps:**

**(i) **\(126^\circ + 44^\circ = 170^\circ\)

As the sum of interior angles on the same side of transversal \(n\) is not \(180^\circ.\)

Therefore, \(l\) is not parallel to \(m.\)

**(ii) Reasoning:**

Let’s visually model this problem. There are two operations that can be done in a sequence. First find the value of \(x\) and then check it is equal to its corresponding angle or not. According to this model, the resultant value of \(x\) is not equal to its corresponding angle. Now, it’s a matter of finding \(l\) is parallel to \(m\) or not

**(ii) ****Steps:**

\[\begin{align}\angle x + 75^\circ &= 180^\circ \text{(Linear pair)}\\\angle x &= 180^\circ - 75^\circ\\\angle x &= 105^\circ\end{align}\]

For \(l\) and \(m\) to be parallel measure of their corresponding angles should be equal but here the measure of \(\angle x\) is \(105^\circ\) and its corresponding angle is \(75^\circ.\)

Therefore, the lines \(l\) and \(m\) are not parallel.

**(iii) Reasoning:**

Let’s visually model this problem. There are two operations that can be done in a sequence. First find the value of \(x\) and then check it is equal to its corresponding angle or not. According to this model, the resultant value of \(x\) is not equal to its corresponding angle. Now, it’s a matter of finding \(l\) is parallel to \(m\) or not

**(iii) Steps:**

\[\begin{align}\angle y&=57^\circ \begin{bmatrix}\text{Vertically opposite}\\\text {angles}\end{bmatrix}\\\angle x + 123^\circ &= 180^\circ\text{ (Linear pair)}\\\angle x& = 180^\circ - 123^\circ\\\angle x &= 57^\circ\end{align}\]

Here, the measure of corresponding angles are equal i.e \(57^\circ.\)

Therefore, lines l and m are parallel to each other.

**(iv) Reasoning:**

Let’s visually model this problem. There are two operations that can be done in a sequence. First find the value of \(x\) by using linear pairand then check it is equal to its corresponding angle or not. According to this model, the resultant value of \(x\) is not equal to its corresponding angle. Now, it’s a matter of finding \(l\) is parallel to \(m\) or not

**(iv) Steps:**

\[\begin{align}\angle x + 98^\circ &= 180^\circ \text{(Linear pair)}\\\angle x &= 180^\circ - 98^\circ\\\angle x &= 82^\circ\end{align}\]

For \(l\) and \(m\) to be parallel measure of their corresponding angles should be equal but here the measure of corresponding angles are \(82^\circ\) and \(72^\circ\) whichn are not equal.

Therefore, \(l\) and \(m\) are not parallel to each other.

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